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| {{Continuum mechanics}}
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| '''Acoustic metamaterials''' are artificially fabricated materials designed to control, direct, and manipulate [[sound wave]]s as these might occur in [[gas]]es, [[liquid]]s, and [[solid]]s. The hereditary line into acoustic [[metamaterial]]s follows from theory and research in [[negative index material]]. Furthermore, with acoustic metamaterials controlling [[sound wave|sonic wave]]s can now be extended to the [[negative refraction]] domain.<ref name=acousticmeta1107/>
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| Control of the various forms of sound [[wave]]s is mostly accomplished through the [[bulk modulus]] ''β'', [[mass density]] ''ρ'', and [[chirality (electromagnetism)|chirality]]. The density and [[bulk modulus]] are analogies of the electromagnetic parameters, [[permittivity]] and [[Permeability (electromagnetism)|permeability]] in negative index materials. Related to this is the mechanics of wave propagation in a [[lattice model (physics)|lattice]] structure. Also materials have [[mass]], and instrinsic degrees of [[stiffness]]. Together these form a [[resonant]] system, and the mechanical (sonic) resonance may be excited by appropriate sonic frequencies (for example [[pulse]]s at audio frequencies).<ref name=acousticmeta1107/>
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| ==History of the acoustic metamaterials==
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| {{main|History of metamaterials}}
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| Acoustic metamaterials have developed from the research and results behind metamaterials. The novel material was originally proposed by [[Victor Veselago]] in 1967, but not realized until some 33 years later. [[John Pendry]] produced the basic elements of metamaterials during the last part of the 1990s. His materials were combined and then negative index materials were realized first in the year 2000 and 2001 which produced a [[negative refraction]] thereby broadening possible optical and material responses. Hence, research in acoustic metamaterials has the same goal of broader material response with sound waves.<ref>Nahin, P.J., Spectrum, IEEE, Volume 29, Issue 3, March 1992 Page(s):45–</ref><ref>{{cite web | title = James Clerk Maxwell | url = http://www.ieeeghn.org/wiki/index.php/James_Clerk_Maxwell |accessdate=2011-06-21 |publisher= IEEE Global History Network |year= 2011}}</ref><ref name=JC-Bose-honored>
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| {{Cite journal| last = D.T.| first =Emerson| title =The work of Jagadis Chandra Bose: 100 years of millimeter-wave research| journal =Microwave Theory and Techniques, IEEE Transactions| volume =45| issue = 12| pages =2267| date =December 1997
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| | format =A facility of the NSF provides added material to the original paper - [http://www.tuc.nrao.edu/~demerson/bose/bose.pdf The work of Jagadish Chandra Bose: 100 years of milmeter wave research].| doi =10.1109/22.643830| postscript = .|bibcode = 1997ITMTT..45.2267E }}</ref><ref name=JC-Bose-paper>
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| {{Cite journal
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| | last =Bose| first =Jagadis Chunder | authorlink =Jagadish Chandra Bose
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| | title =On the Rotation of Plane of Polarisation of Electric Waves by a Twisted Structure
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| | journal =Proceedings of the Royal Society
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| | volume = 63| issue =1| pages =146–152
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| | date =1898-01-01
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| | doi =10.1098/rspl.1898.0019
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| | postscript =.
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| }}</ref><ref name=physicsengineering1/><ref name=physicsengineering1>
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| {{cite book|last = Nader|first = Engheta|coauthors = Richard W. Ziolkowski
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| |title = Metamaterials: physics and engineering explorations|publisher = Wiley & Sons
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| |date = June 2006|pages = xv
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| |url = http://books.google.com/?id=51e0UkEuBP4C|isbn = 978-0-471-76102-0}}</ref><ref name=MTM-lecture>
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| {{Cite journal| last = Engheta| first = Nader| title =Metamaterials| journal =[[U Penn]] Dept. of Elec. and Sys. Engineering| volume =and Workshop| pages =99| series = Lecture| date =2004-04-29
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| | url =http://radar04.lightsky.net/workshops/engheta/metamaterials.pdf
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| | format =Nader Engheta co-authored [http://books.google.com/books?id=51e0UkEuBP4C&pg=PP1&dq=metamaterials#v=onepage&q=&f=false Metamaterials: physics and engineering explorations].| id =| postscript = . }}</ref><ref>{{cite journal | last1 = Shelby | first1 = R. A. | last2 = Smith | first2 = D. R. | last3 = Schultz | first3 = S. | year = 2001 | title = Experimental verification of a negative index of refraction | url = | journal = Science | volume = 292 | issue = 5514| pages = 77–79 |bibcode = 2001Sci...292...77S |doi = 10.1126/science.1058847 | pmid = 11292865 }}</ref><ref>Negative refraction (electromagnetic) first demonstrated by D. Smith, S. Shultz, and R. Shelby (2000–2001)</ref>
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| Research employing acoustic metamaterials began in the year 2000 with the fabrication and demonstration of sonic crystals in a liquid.<ref name=soniccrystals/> This was followed by transposing the behavior of the split-ring resonator to research in acoustic metamaterials.<ref name=comp/> After this double negative parameters (negative bulk modulus ''β''<sub>eff</sub> and negative density ''ρ''<sub>eff</sub>) were produced by this type of medium.<ref name=acousticdbleneg/> Then a group of researchers presented the design and tested results of an ultrasonic metamaterial lens for focusing 60 kHz.<ref name=underwaterlens09/>
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| The earlier studies of acoustics in technology, which is called [[acoustical engineering]], are typically concerned with how to reduce unwanted sounds, [[noise control]], how to make useful sounds for the medical diagnosis, [[sonar]], and [[sound reproduction]] and how to measure some other physical properties using sound.
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| Using acoustic metamaterials the directions of sound through the medium can be controlled by manipulating the [[refractive index]]. Therefore the traditional acoustic technologies are extended and may eventually cloak certain objects from acoustic detection.
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| ==Basic principles==
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| Since the acoustic metamaterials are one of the branch of the [[metamaterials]], the basic principle of the acoustic metamaterials is similar to the principle of metamaterials. These metamaterials usually gain their properties from structure rather than composition, using the inclusion of small inhomogeneities to enact effective macroscopic behavior.<ref name=physicsengineering1>{{cite book|last = Nader|first = Engheta|coauthors = Richard W. Ziolkowski
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| |title = Metamaterials: physics and engineering explorations|publisher = Wiley & Sons
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| |date = June 2006|pages = xv, 3–30, 37, 215–233, 240, 241
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| |url = http://books.google.com/?id=51e0UkEuBP4C|isbn = 978-0-471-76102-0}}</ref><ref name=smithmetamaterials1>{{cite web
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| |last = Smith|first = David R.| authorlink = David R. Smith|title = What are Electromagnetic Metamaterials?
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| |work = Novel Electromagnetic Materials|publisher = The research group of D.R. Smith
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| |date = 2006-06-10|url = http://people.ee.duke.edu/~drsmith/about_metamaterials.html
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| |accessdate = 2009-08-19}}</ref>
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| Similar to metamaterials research, investigating materials with [[Negative index metamaterials]], the negative index acoustic metamaterials became the primary research. Negative refractive index of acoustic materials can be achieved by changing the [[bulk modulus]] and [[mass density]].
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| ===Bulk modulus and mass density===
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| Below, the [[bulk modulus]] ''β'' of a substance reflects the substance's resistance to uniform compression. It is defined in relation to the [[pressure]] increase needed to cause a given relative decrease in volume.
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| The [[Density|mass density]] (or just "density") of a material is defined as mass per unit volume and is expressed in grams per cubic centimeter (g/cm<sup>3</sup>).<ref name=densitygroiler>{{cite encyclopedia|title =Density| encyclopedia = Encyclopedia Americana|volume = Online|publisher =Scholastic Inc| year =2009|series =Grolier|url =http://ea.grolier.com/article?id=0124170-00| accessdate =2009-09-06}}</ref> In all three classic states of matter — gas, liquid, or [[solid]] — the density varies with a change in temperature or pressure, and [[gas]]es are the most susceptible to those changes. The spectrum of densities is wide ranging: from 10<sup>15</sup> g/cm<sup>3</sup> for [[neutron star]]s, 1.00 g/cm<sup>3</sup> for water to 1.2×10<sup>−3</sup> g/cm<sup>3</sup> for air.<ref name=densitygroiler/> Also relevant here are [[area density]] which is mass over a (two-dimensional) area, [[linear density]] - mass over a one-dimensional line, and [[relative density]], which is a density divided by the density of a reference material, such as water.
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| For acoustic materials and acoustic metamaterials, both bulk modulus and density are component parameters, which define their [[refractive index]].
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| ===Acoustic metamaterial analogues===
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| [[File:Isostatic pressure deformation.png|thumb|right|Bulk modulus - illustration of uniform compression]]
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| Scientific research revealed that acoustic metamaterials have analogues to electromagnetic metamaterials when exhibiting the following characteristics:
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| In certain [[frequency band]]s, the ''[[density|effective mass density]]'' and ''[[bulk modulus]]'' may become negative. This results in a ''negative refractive index''. ''Flat slab focusing'', which can result in ''[[super resolution]]'', is similar to electromagnetic metamaterials. The double negative parameters are a result of low-frequency [[resonance]]s.<ref name=acousticmeta1107/> In combination with a well-defined [[polarization (waves)|polarization]] during wave propagation; '''''k''''' ''= |n|ω'', is an equation for refractive index as sound waves interact with acoustic metamaterials (below):<ref name=acousticbook1>{{cite book|last =Krowne| first = Clifford M.| coauthors = Yong Zhang
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| |title = Physics of Negative Refraction and Negative Index Materials: Optical and Electronic Aspects and Diversified Approaches|publisher =Springer-Verlag| year =2007
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| |location =New York|page=183 (Chapter 8)
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| |url = http://books.google.com/?id=AjdbPVfK7rkC&pg=PA183|isbn = 978-3-540-72131-4}}</ref>
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| ::<math>n^2=\frac{\rho}{\beta} </math>
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| The inherent parameters of the medium are the mass density ''ρ'', bulk modulus β, and chirality '''''k'''''. Chirality, or handedness, determines the polarity of [[wave propagation]] ([[wave vector]]). Hence within the last equation, Veselago-type solutions (n<sup>2</sup> = ''u''*ε) are possible for wave propagation as the negative or positive state of ''ρ'' and β determine the forward or backward wave propagation.<ref name=acousticbook1/>
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| In negative refractive, electromagnetic metamaterials, negative permittivity can be found in natural materials. However, negative permeability has to be intentionally created in the [[Transmission medium|artificial transmission medium]]. Obtaining a negative refractive index with acoustic materials is different.<ref name=acousticbook1/>
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| Neither negative ''ρ'' nor negative β are found in naturally occurring materials;<ref name=acousticbook1/>
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| they are derived from the [[resonant|resonant frequencies]] of an artificially fabricated transmission medium (metamaterial), and such negative values are an anomalous response. Negative ''ρ'' or β means that at certain frequencies the medium expands when experiencing [[Gain compression|compression]] (negative modulus), and accelerates to the left when being pushed to the right (negative density).<ref name=acousticbook1/>
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| ===Electromagnetic field vs acoustic field===
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| The electromagnetic spectrum extends from below frequencies used for modern radio to [[Gamma ray|gamma radiation]] at the short-wavelength end, covering wavelengths from thousands of kilometers down to a fraction of the size of an atom. That would be wavelengths from 10<sup>3</sup> to 10<sup>−15</sup> kilometers. The long wavelength limit is the size of the universe itself, while it is thought that the short wavelength limit is in the vicinity of the Planck length, although in principle the spectrum is infinite and continuous.
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| Infrasonic frequencies range from 20 Hz down to 0.001 Hz. Audible frequencies are 20 Hz to 20 kHz. Ultrasonic range is above 20 kHz. Sound requires a medium. Electromagnetics radiation (EM waves) can travel in a vacuum.
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| ===Mechanics of lattice waves===
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| [[File:Lattice wave.svg|200px|right]]
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| An imaginary demonstration: A hypothetical rigid [[Crystal structure|lattice]] structure (solid) is composed of 10<sup>23</sup> atoms. However, in a real solid these particles could just as easily be [[ions]]. In a rigid lattice structure, atoms exert pressure, or a force, on each other in order to maintain [[wikt:equilibrium|equilibrium]]. Atomic forces maintain rigid lattice structure. Most of them, such as the [[covalent bond|covalent]] or [[ionic bond]]s, are of electric nature. The [[magnetic force]], and the force of [[gravity]] are negligible.<ref name=latticemechanics2>{{cite book| last =Lavis| first =David Anthony
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| |coauthors =George Macdonald Bell
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| |title = Statistical Mechanics Of Lattice Systems. Volume 2
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| |publisher = Springer-Verlag|year = 1999| location = New York
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| |pages = 1–4|url=http://books.google.com/?id=2ztrXFKT2c0C&pg=PA1|isbn =978-3-540-64436-1}}</ref>
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| Because of bonding between [[atom]]s, the displacement of one or more atoms from their equilibrium positions will give rise to a set of vibration [[wave]]s propagating through the [[Crystal structure|lattice]]. One such wave is shown in the figure to the right. The [[amplitude]] of the wave is given by the displacements of the atoms from their equilibrium positions. The [[wavelength]] λ is marked.<ref name=latticemechanics1>{{cite book| last = Brulin| first =Olof| coauthors =Richard Kin Tchang Hsieh
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| |title = Mechanics of micropolar media| publisher = World Scientific Publishing Company
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| |year = 1982|pages= 3–11
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| |url = http://books.google.com/?id=6S4rbmb1IqUC&pg=PA3|isbn =9971-950-02-2}}</ref>
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| There is a ''minimum possible'' wavelength, given by the equilibrium separation ''a'' between atoms. Any wavelength shorter than this can be mapped onto a wavelength longer than ''a'', due to effects similar to that in [[aliasing]].<ref name=latticemechanics1/>{{-}}
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| ==Acoustic metamaterials analysis and experiments==
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| The current research on acoustic metamaterials is based not only on prior experience with [[Metamaterial|electromagnetic metamaterials]]. The key [[physics]] in [[acoustics]] are ''sound'', ''[[ultrasound]]'' and ''[[infrasound]]'', which are [[elastic waves|mechanical waves]] in [[gas]]es, [[liquid]]s, and solids. One objective of the inquiry into the properties of acoustic metamaterials is applications in [[seismic wave]] reflection and in [[vibration]] control technologies related to [[earthquake]]s.<ref name=acousticmeta1107/><ref name=soniccrystals/><ref name=srr/>
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| ===Sonic crystals===
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| In the year 2000 the research of Liu ''et al.'' paved the way to acoustic metamaterials through sonic [[crystal]]s. The latter exhibit spectral gaps two orders of magnitude smaller than the wavelength of sound. The spectral gaps prevent the transmission of waves at prescribed frequencies. The frequency can be tuned to desired parameters by varying the size and geometry of the metamaterial.<ref name=soniccrystals>{{cite journal
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| |last = Zhengyou Liu| first = Liu
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| |coauthors = Xixiang Zhang, Yiwei Mao, Y. Y. Zhu,Zhiyu Yang, C. T. Chan, Ping Sheng
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| |title = Locally Resonant Sonic Materials|journal=Science|volume = 289
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| |issue = 5485|pages=1734–1736| year = 2000|doi = 10.1126/science.289.5485.1734
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| |pmid = 10976063
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| |postscript = .|bibcode = 2000Sci...289.1734L }}</ref>
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| The fabricated material consisted of a high-density solid lead ball as the core, one centimeter in size, which was coated with a 2.5-mm layer of rubber [[silicone]]. These were arranged in a crystal lattice structure of an 8 × 8 × 8 cube. The balls were cemented into the cubic structure with an [[epoxy]]. Transmission was measured as a function of frequency from 250 to 1600 Hz for effectively a four-layer sonic crystal. A two-centimeter slab absorbed sound that normally would require a much thicker material, at 400 Hz. A drop in amplitude was observed at 400 and 1100 Hz.<ref name=soniccrystals/><ref name=iopsoniccrystals>{{Cite book
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| |publisher = [[Institute of Physics]]
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| |title = Sonic crystals make the sound barrier
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| |date =2000-09-07
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| |url = http://physicsworld.com/cws/article/news/2787
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| |accessdate = 2009-08-25}}</ref>
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| The amplitudes of the sound waves entering the surface were compared with the sound waves at the center of the metamaterial structure. The oscillations of the coated spheres absorbed sonic energy, which created the frequency gap; the sound energy is absorbed exponentially as the thickness of the material is increased. The key result here is a negative elastic constant created from resonant frequencies of the material. Its projected applications, with a future expanded frequency range in elastic wave systems, are seismic wave reflection and ultrasonics.<ref name=soniccrystals/><ref name=iopsoniccrystals/>
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| ===Split-ring resonators for acoustic metamaterials===
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| [[File:Left-handed metamaterial array configuration.jpg|thumb|250px|Copper split-ring resonators and wires mounted on interlocking sheets of fiberglass circuit board. A split-ring resonator consists of an inner square with a split on one side embedded in an outer square with a split on the other side. The split-ring resonators are on the front and right surfaces of the square grid and the single vertical wires are on the back and left surfaces.<ref name=comp>
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| {{cite journal|url=http://people.ee.duke.edu/~drsmith/pubs_smith_group/Smith_PRL_84_4184_(2000).pdf|doi=10.1103/PhysRevLett.84.4184|title=Composite Medium with Simultaneously Negative Permeability and Permittivity|year=2000|last1=Smith|first1=D. R.|journal=Physical Review Letters|volume=84|pages=4184–7|pmid=10990641|last2=Padilla|first2=WJ|last3=Vier|first3=DC|last4=Nemat-Nasser|first4=SC|last5=Schultz|first5=S|issue=18|postscript=.|bibcode=2000PhRvL..84.4184S}}</ref><ref name=shelby>{{cite journal|doi=10.1063/1.1343489|title=Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial|year=2001|last1=Shelby|first1=R. A.|journal=Applied Physics Letters|volume=78|issue=4|pages=489|last2=Smith|first2=D. R.|last3=Nemat-Nasser|first3=S. C.|last4=Schultz|first4=S.|postscript=.|bibcode = 2001ApPhL..78..489S }}</ref>]]
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| In 2004 [[split-ring resonator]]s (SRR) became the object of acoustic metamaterial research.<ref name=srr>{{cite journal| last = Movchan| first = A. B.
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| |coauthors = and S. Guenneau| title = Split-ring resonators and localized modes
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| |journal =Phys. Rev. B| volume = 70|issue = 12|page=125116| year = 2004|url = http://www.maths.liv.ac.uk/~guenneau/prb2004a.pdf
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| |doi =10.1103/PhysRevB.70.125116|accessdate =2009-08-27| postscript = .|bibcode = 2004PhRvB..70l5116M }}</ref>
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| Prior research with SRRs fabricated as negative index ''electromagnetic'' metamaterials was referenced as the progenitor of further research in acoustic metamaterials.<ref name=srr/>
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| An analysis of the frequency band gap characteristics, derived from the inherent [[Asymptotic analysis|limiting properties]] of artificially created SRRs, paralleled an analysis of sonic crystals. The band gap properties of SRRs were related to sonic crystal band gap properties.<ref name=srr/> Inherent in this inquiry is a description of [[List of materials properties#Mechanical properties|mechanical properties]] and problems
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| of [[continuum mechanics]] for sonic crystals, as a macroscopically homogeneous substance.<ref name=srr/>
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| The correlation in [[bandgap]] capabilities includes locally resonant elements and [[elastic moduli]] which operate in a certain [[frequency range]]. Elements which interact and resonate in their respective localized area are embedded throughout the material. In acoustic metamaterials, locally resonant elements would be the interaction of a single 1-cm rubber sphere with the surrounding liquid. The values of the stop band and band gap frequencies can be controlled by choosing the size, types of materials, and the integration of microscopic structures which control the modulation of the frequencies. These materials are then able to shield [[acoustics|acoustic]] signals and attenuate the effects of anti-plane shear waves. By extrapolating these properties to larger scales it could be possible to create seismic wave filters (see [[Metamaterial#Seismic metamaterials|Seismic metamaterials]]).<ref name=srr/>
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| According to [[research]] prior to this analysis, arrayed metamaterials can create filters or [[polarizer]]s of either [[Electromagnetic radiation|electromagnetic]] or [[elastic wave]]s.<ref name=srr/>
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| Here a method is shown which can be applied to two-dimensional [[stop band]] and [[bandgap]] control with either photonic or sonic structures.<ref name=srr/>
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| Similar to [[photonic]] and electromagnetic metamaterial fabrication, a sonic metamaterial is embedded with localized sources of mass density ''ρ'' and the (elastic) [[bulk modulus]] β parameters, which are analogous to permittivity and permeability, respectively. The sonic (or phononic) metamaterials are [[sonic crystals]], as in the previous section. These crystals have a solid [[lead]] core and a softer, more elastic [[silicone]] coating.<ref name=soniccrystals/> The sonic crystals had built-in localized resonances due to the coated spheres which resulted in almost flat [[Acoustic dispersion|dispersion]] curves. Low-frequency bandgaps and localized wave interactions of the coated spheres were analyzed and presented in.<ref name=srr/>
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| This method can be used to tune bandgaps inherent in the material and, also, create new low-frequency bandgaps. It is also applicable for designing low-frequency phononic crystal waveguides ([[radio frequency]]). Doubly [[periodic function|periodic]] square array of SRRs are used to illustrate the methodology.<ref name=srr/>
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| ===Phononic crystal===
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| '''Phononic crystals''' are synthetic materials that are formed by [[split-ring resonator|periodic variation]] of the [[acoustics|acoustic properties]] of the material (i.e., [[elasticity (physics)|elasticity]] and [[mass]]). One of the main properties of the phononic crystals is the possibility of having a phononic [[bandgap]]. A phononic crystal with phononic bandgap prevents [[phonons]] of selected ranges of [[frequencies]] from being transmitted through the [[transmission medium|material]].<ref name=Phononic-crystal-1/><ref name=Physics-of-Phonons/>
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| To obtain the frequency band structure of a phononic crystal, Bloch theory is applied on a single unit cell in the reciprocal lattice space (Brillouin zone). Several numerical methods are available for this problem, e.g., the planewave expansion method, the finite element method, and the finite difference method. A brief survey of numerical methods for calculating the frequency band structure is provided by Hussein (2009)<ref name=Hussein2009>{{Cite journal | author = M.I. Hussein| year = 2009 | title = Reduced Bloch mode expansion for periodic media band structure calculations | journal = Proceedings of the Royal Society A | volume = 465 | issue = 2109 | pages = 2825–2848 | doi = 10.1098/rspa.2008.0471 | postscript = . |bibcode = 2009RSPSA.465.2825H |arxiv = 0807.2612 }}</ref>
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| In order to speed up the calculation of the frequency band structure, the '''Reduced Bloch Mode Expansion (RBME)''' method can be used.<ref name="Hussein2009"/> The RBME applies "on top" of any of the primary expansion numerical methods mentioned above. For large unit cell models, the RBME method can reduce the time for computing the band structure by up to two orders of magnitude.
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| The basis of phononic crystals dates back to [[Isaac Newton]] who imagined that [[sound waves]] [[wave propagation|propagated]] through [[air]] in the same way that an [[elastic wave]] would propagate along a [[lattice model (physics)|lattice]] of [[point mass]]es connected by springs with an [[Elasticity (physics)|elastic force]] constant E. This [[force]] constant is identical to the [[Bulk modulus|modulus]] of the [[silicon|material]]. Of course with phononic crystals of [[Fluid mechanics|materials]] with differing modulus the [[Continuum mechanics|calculations]] are a little more complicated than this simple model.<ref name=Phononic-crystal-1/><ref name=Physics-of-Phonons/>
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| Based on Newton’s observation we can conclude that a key factor for [[Earthquake engineering|acoustic band-gap engineering]] is [[Acoustic impedance|impedance]] mismatch between [[periodic function|periodic]] elements comprising the crystal and the surrounding medium. When an [[longitudinal wave|advancing wave-front]] meets a material with very high impedance it will tend to increase its [[phase velocity]] through that medium. Likewise, when the advancing wave-front meets a low impedance medium it will slow down. We can exploit this concept with periodic (and handcrafted) arrangements of impedance mismatched elements to affect [[sound waves|acoustic waves]] in the crystal – essentially band-gap [[engineering]].<ref name=Phononic-crystal-1/><ref name=Physics-of-Phonons/>
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| The position of the band-gap in frequency space for a phononic crystal is controlled by the size and arrangement of the elements comprising the crystal. The width of the band gap is generally related to the difference in the [[speed of sound]] (due to impedance differences) through the materials that comprise the composite.<ref name=Phononic-crystal-1>
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| {{Cite news| last = Gorishnyy| first =Taras| coauthors =Martin Maldovan, Chaitanya Ullal and Edwin Thomas| title =Sound ideas| newspaper =Physicsworld.com| publisher =[[Institute of Physics]]| date =2005-12-01| url =http://physicsworld.com/cws/article/print/23671
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| | accessdate =2009-11-05 }}</ref><ref name=Physics-of-Phonons>
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| {{cite book |title=The Physics of Phonons |author=G.P Srivastava |isbn=0-85274-153-7 |year=1990 |publisher=CRC Press}}</ref>
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| ===Double-negative acoustic metamaterial===
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| [[File:Sine waves same phase.svg|thumb|In-phase waves]]
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| [[File:Sine waves different phase.svg|thumb|Out-of-phase waves]]
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| [[File:Phase-shift illustration.png|right|thumb|Left: the [[real part]] of a [[plane wave]] moving from top to bottom. Right: the same wave after a central section underwent a phase shift, for example, by passing through metamaterial [[Homogeneity (physics)|inhomogeneities]] of different thickness than the other parts. (The illustration on the right ignores the effect of [[diffraction]] whose effect increases over large distances).]]
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| The electromagnetic (isotropic) metamaterials have built-in [[resonance]] structures that exhibit effective negative permittivity and negative permeability for some frequency ranges. In contrast, it is difficult to build composite [[acoustics|acoustic materials]] with built-in resonances such that the two effective [[response function]]s are negative within the capability or range of the [[transmission medium]].<ref name=acousticdbleneg>{{cite journal| last = Li| first = Jensen| coauthors = C. T. Chan| title = Double-negative acoustic metamaterial| journal = Phys. Rev. E| volume =70|issue =5|doi = 10.1103/PhysRevE.70.055602| year = 2004| pages = 055602| postscript = .|bibcode = 2004PhRvE..70e5602L }}</ref>
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| The mass density ''ρ'' and bulk modulus β are position dependent. Using the formulation of a [[plane wave]] the wave vector is:<ref name=acousticdbleneg/>
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| ::::::<math> \vec{k} = \frac{\ |n| \omega}{c}. \,</math>
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| The [[angular frequency]] is represented by ''ω'' and ''c'' is the propagation speed of acoustic signal through the [[homogeneity (physics)|homogeneous medium]]. With constant density and bulk modulus as constituents of the medium, the refractive index is expressed as n<sup>2</sup> = ''ρ'' / β. In order to develop a propagating (plane) wave through the material, it is necessary for both ''ρ'' and β to be either positive or negative.<ref name=acousticdbleneg/>
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| When the negative parameters are achieved, the mathematical result of the [[Poynting vector]] <math> \scriptstyle \overleftarrow{s} </math>. is the opposite direction of the [[wave vector]] <math> \scriptstyle \overrightarrow{k}</math>. This requires negativity in bulk modulus and density. Physically, it means that the medium displays an anomalous response at some frequencies such that it expands upon compression (negative bulk modulus) and moves to the left when being pushed to the right (negative density) at the same time.<ref name=acousticdbleneg/>
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| Natural materials do not have a negative density or a negative bulk modulus, but, negative values are mathematically possible, and can be demonstrated when dispersing soft rubber in a liquid.<ref name=acousticdbleneg/><ref name=tapwater>{{cite book| last =Trostmann|first =Erik| title =Tap water as a hydraulic pressure medium| publisher =CRC Press|date =200-11|pages =36|url =http://books.google.com/?id=EiaCkItHNRAC&pg=PA46|isbn =978-0-8247-0505-3}}</ref><ref name=fluidmechbasic>{{cite book| last=Petrila|first = Titus| coauthors = Damian Trif|title =Basics of fluid mechanics and introduction to computational fluid dynamics|publisher =Springer-Verlag New York, LLC| date =December 2004|url =http://books.google.com/?id=pMH_fEFdilMC&printsec=frontcover|isbn =978-0-387-23837-1}}</ref>
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| Even for composite materials, the effective bulk modulus and density should be normally bounded by the values of the constituents, i.e., the derivation of lower and upper bounds for the elastic moduli of the medium. Intrinsic is the expectation for positive bulk modulus and positive density. For example, dispersing spherical solid particles in a fluid results in the ratio governed by the specific gravity when interacting with the long acoustic wavelength (sound). Mathematically, it can be easily proven that β<sub>eff</sub> and ''ρ''<sub>eff</sub> are definitely positive for natural materials.<ref name=acousticdbleneg/><ref name=tapwater/> The exception occurs at low resonant frequencies.<ref name=acousticdbleneg/>
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| As an example, acoustic double negativity is theoretically demonstrated with a composite of soft, silicone rubber spheres suspended in water.<ref name=acousticdbleneg/> In soft rubber, sound travels much slower than through the water. The high velocity contrast of sound speeds between the rubber spheres and the water allows for the transmission of very low monopolar and dipolar frequencies. This is an analogue to analytical solution for the scattering of electromagnetic radiation, or [[Mie theory|electromagnetic plane wave scattering]], by spherical particles - [[dielectric]] spheres.<ref name=acousticdbleneg/>
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| Hence, there is a narrow range of normalized frequency 0.035 < ωa/(2πc) < 0.04 where the bulk modulus and negative density are both negative. Here ''a'' is the lattice constant if the spheres are arranged in a [[Cubic crystal system|face-centered cubic]] (fcc) lattice; ω is frequency and ''c'' is speed of the acoustic signal. The effective bulk modulus and density near the static limit are positive as predicted. The monopolar resonance creates a negative bulk modulus above the normalized frequency at about 0.035 while the dipolar resonance creates a negative density above the normalized frequency at about 0.04.<ref name=acousticdbleneg/>
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| This behavior is analogous to low-frequency resonances produced in SRRs (electromagnetic metamaterial). The wires and split rings create intrinsic electric dipolar and magnetic dipolar response. With this artificially constructed acoustic metamaterial of rubber spheres and water, only one structure (instead of two) creates the low-frequency resonances to achieve double negativity.<ref name=acousticdbleneg/> With monopolar resonance, the spheres expand, which produces a phase shift between the waves passing through rubber and water. This creates the negative response. The dipolar resonance creates a negative response such that the frequency of the center of mass of the spheres is out of phase with the wave vector of the sound wave (acoustic signal). If these negative responses are large enough to compensate the background fluid, one can have both negative effective bulk modulus and negative effective density.<ref name=acousticdbleneg/>
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| Both the mass density and the reciprocal of the bulk modulus are decreasing in magnitude fast enough so that the group velocity becomes negative (double negativity). This gives rise to the desired results of negative refraction. The double negativity is a consequence of resonance
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| and the resulting negative refraction properties.
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| ===Metamaterial with simultaneously negative bulk modulus and mass density===
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| In August 2007 a metamaterial was reported which simultaneously possesses a negative bulk modulus and mass density. This metamaterial is a [[Cubic crystal system|zinc blende]] structure consisting of one [[Cubic crystal system|fcc]] array of bubble-contained-water spheres (BWSs) and another relatively shifted fcc array of rubber-coated-gold spheres (RGSs) in special epoxy.<ref name=simultaneousnegative>{{cite journal| last = Ding| first = Yiqun| coauthors = et al.|title = Metamaterial with Simultaneously Negative Bulk Modulus and Mass Density| journal = Phys. Rev. Lett.| volume =99|issue = 9|doi = 10.1103/PhysRevLett.99.093904|year = 2007| pages = 093904| pmid = 17931008| postscript = .| bibcode=2007PhRvL..99i3904D}}</ref>
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| Negative bulk modulus is achieved through monopolar resonances of the BWS series. Negative mass density is achieved with dipolar resonances of the gold sphere series. Rather than rubber spheres in liquid, this is a solid based material. This is also as yet a realization of simultaneously negative bulk modulus and mass density in a solid based material, which is an important distinction.<ref name=simultaneousnegative/>
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| {{Further|Poisson's ratio}}
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| ===Double C resonators===
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| Double C resonator (DCR) is a ring cut in halves. In 2007, proposals have been made for arrays of DCRs and similar negative acoustic metamaterial.<ref name=acousticmeta1107>{{cite journal|last =Guenneau|first =Sébastien
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| |coauthors =Alexander Movchan, Gunnar Pétursson, and S. Anantha Ramakrishna
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| |title =Acoustic metamaterials for sound focusing and confinement
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| |journal =New Journal of Physics| volume =9| issue =399| pages =1367–2630|year =2007 |doi =10.1088/1367-2630/9/11/399|postscript =.|bibcode = 2007NJPh....9..399G }}</ref> Although linear elasticity is mentioned, the problem is defined around shear waves directed at angles to the plane of the cylinders. The DCR was constructed similar to the SRRs in a multiple cell configuration. The DCR has been improved with stiffer material sheets.<ref name=acousticmeta1107/> Each cell consists of a large rigid disk and two
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| thin ligaments.<ref name=acousticmeta1107/> The DCR cell is a tiny oscillator connected by springs. One spring of the oscillator connects to the ''mass'' and is anchored by the other spring. The [[LC circuit|LC resonator]] has specified capacitance and inductance.<ref name=acousticmeta1107/> The limitations are expressed with appropriate mathematical equations. In addition to the intended limitations is that the speed of sound in the matrix is expressed as c = √''ρ''/µ with a matrix of density ''ρ'' and shear modulus μ. The resonant frequency is then expressed as √1/(LC).<ref name=acousticmeta1107/>
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| A phononic bandgap occurs in association with the resonance of the split cylinder ring. There is a phononic band gap within a range of ''normalized frequencies''. This is when the inclusion moves as a [[rigid body]].
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| The DCR design produced a suitable band with negative slope in a range of frequencies. This band was obtained by hybridizing the modes of a DCR with the modes of thin stiff bars. Calculations have shown that at these frequencies:
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| *a beam of sound negatively refracts across a slab of such a medium,
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| *the phase vector in the medium possesses real and imaginary parts with opposite signs,
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| *the medium is well impedance-matched with the surrounding medium,
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| *a flat slab of the metamaterial can image a source across the slab like a Veselago lens,
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| *the image formed by the flat slab has considerable sub-wavelength image resolution, and
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| *a double corner of the metamaterial can act as an open resonator for sound.
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| ===Acoustic metamaterial superlens===
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| In May 2009 Shu Zhang ''et al.'' presented the design and test results of an ultrasonic metamaterial lens for focusing 60 kHz (~2 cm wavelength) sound waves under water.<ref name=underwaterlens09>{{cite journal|last = Thomas| first = Jessica| title = Metamaterial brings sound into focus
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| |journal =Physics|format=synopsis for Phys. Rev. Lett. 102, 194301|publisher =American Physical Society |date = 2009-05-15
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| |url =http://physics.aps.org/synopsis-for/10.1103/PhysRevLett.102.194301
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| |accessdate = 2009-08-29|doi = 10.1103/PhysRevLett.102.194301|volume = 102|pages = 194301|last2 = Yin|first2 = Leilei|last3 = Fang|first3 = Nicholas|postscript = .|pmid=19518957|issue=19|bibcode=2009PhRvL.102s4301Z}}</ref> The lens is made of sub-wavelength elements and is therefore potentially more compact than phononic lenses that operate in the same frequency range.<ref name=underwaterlens09/>
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| High-resolution acoustic imaging techniques are the essential tools for nondestructive testing and medical screening. However, the spatial resolution of the conventional acoustic
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| imaging methods is restricted by the incident wavelength of ultrasound. This is due to the quickly fading [[Evanescent wave|evanescent fields]] which carry the sub-wavelength features of objects.<ref name=acousticsuperlens>{{cite journal
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| |last = Zhang| first = Shu| coauthors =Leilei Yin and Nicholas Fang
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| |title = Focusing Ultrasound with Acoustic Metamaterial Network| journal =Phys. Rev. Lett.
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| |volume = 102| issue = 19|publisher = American Physical Society|year = 2009
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| |arxiv = 0903.5101|doi = 10.1103/PhysRevLett.102.194301
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| |pages = 194301
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| |pmid = 19518957
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| |postscript = .| bibcode=2009PhRvL.102s4301Z}}</ref>
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| The lens consists of a network of fluid-filled cavities called [[Helmholtz resonance|Helmholtz resonators]] that oscillate at certain sonic frequencies. Similar to a network of inductors and capacitors in electromagnetic metamaterial, the arrangement of Helmholtz cavities designed by Zhang ''et al.'' have a negative dynamic modulus for ultrasound waves. Zhang ''et al.'' did focus a point source of 60.5 kHz sound to a spot size that is roughly the width of half a wavelength and their design may allow to push the spatial resolution even further.<ref name=underwaterlens09/> This result is in excellent agreement with the numerical simulation by transmission line model, which derived the effective mass density and compressibility. This metamaterial lens also displays variable focal length at different frequencies.<ref name=acousticsuperlens/><ref name=acousticsuperlensAdler>{{cite news|last = Adler|first = Robert|last2 = ''Acoustic metamaterials.''|first2 = ''Negative refraction. Earthquake protection.''|title = Acoustic 'superlens' could mean finer ultrasound scans|magazine = New Scientist Tech|page = 1|year = 2008|date = 01-08|url =http://www.newscientist.com/article/dn13156-acoustic-superlens-could-mean-finer-ultrasound-scans.html|accessdate =2009-08-12}}</ref>
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| ===Acoustic diode===
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| An acoustic diode was introduced in August 2009. An electrical [[diode]] allows current to flow in only one direction in a wire; it is an essential electronic device which had no analogues for sound waves. However, the reported design partially fills this role by converting sound to a new frequency and blocking any backwards flow of the original frequency. In practice, it could give designers new flexibility in making ultrasonic sources like those used in medical imaging. The proposed structure combines two components: The first is a sheet of nonlinear acoustic material—one whose sound speed varies with air pressure. An example of such a material is a collection of grains or beads, which becomes stiffer as it is squeezed. The second component is a filter that allows the doubled frequency to pass through but reflects the original.<ref name=acousticdiode>{{cite journal| last = Monroe| first =Don|title = One-way Mirror for Sound Waves|journal =Physical Review Focus
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| |format=synopsis for "Acoustic Diode: Rectification of Acoustic Energy Flux in One-Dimensional Systems" by Bin Liang, Bo Yuan, and Jian-chun Cheng|publisher =American Physical Society|date =2009-08-25| url =http://focus.aps.org/story/v24/st8#author| accessdate =2009-08-28| postscript = .}}</ref><ref>{{cite journal|doi=10.1103/PhysRevLett.93.184301|title=Thermal Diode: Rectification of Heat Flux|year=2004|last1=Li|first1=Baowen|journal=Physical Review Letters|volume=93|pages=184301|pmid=15525165|last2=Wang|first2=L|last3=Casati|first3=G|issue=18|postscript=.|bibcode=2004PhRvL..93r4301L|arxiv = cond-mat/0407093 }}</ref>
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| ===Acoustic cloaking===
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| {{main|Metamaterial cloaking}}
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| An '''acoustic cloak''' is a hypothetical device that would make objects impervious towards sound waves. This could be used to build [[sound proof]] homes, advanced concert halls, or stealth warships. The mathematics and physics behind acoustic cloaking has been known for several years. The idea of acoustic cloaking is to deviate the sounds waves around the object that has to be cloaked. But realizing it in materials has been difficult, since [[mechanical Metamaterials]] are needed. The key to this problem is acoustic metamaterials also known as "Sonic Crystals." Making a metamaterial for sound means identifying the acoustic analogues to permittivity and permeability in light waves. It turns out that these are the material's mass density and its elastic constant. Researchers from [[Wuhan University]], [[China]] in a paper<ref>{{cite journal |doi=10.1103/PhysRevLett.99.093904 |title=Metamaterial with Simultaneously Negative Bulk Modulus and Mass Density |year=2007 |last1=Ding |first1=Yiqun |last2=Liu |first2=Zhengyou |last3=Qiu |first3=Chunyin |last4=Shi |first4=Jing |journal=Physical Review Letters |volume=99 |issue=9 |pmid=17931008|bibcode = 2007PhRvL..99i3904D |pages=093904 }}</ref> in 2007 reported such a metamaterial which simultaneously possessed a negative bulk modulus and mass density.
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| ====Potential applications====
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| If such a material could be commercialized, researchers believe it could have many applications. Walls of the material could be built to soundproof houses or it could be used in concert halls to enhance acoustics or direct noise away from certain areas. The military may also be interested to conceal submarines from detection by sonar or to create a new class of stealth ships.
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| ===Metamaterial acoustic cloak===
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| A laboratory metamaterial device that is applicable to [[ultra-sound]] waves has been demonstrated in January 2011. It can be applied to sound [[wavelengths]] from 40 to 80 kHz.
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| The metamaterial acoustic cloak is designed to hide objects sumberged in water. The metamaterial cloaking mechanism bends and twists sound waves by intentional design.
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| The cloaking mechanism consists of 16 [[concentric rings]] in a cylindrical configuration, and each ring with acoustic circuits. It is intentionally designed to guide sound waves, in two dimensions. The first [[microwave metamaterial cloak]] guided electromagnetic waves in two dimensions.
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| Each ring has a different [[index of refraction]]. This causes sound waves to vary their speed from ring to ring. "''The [[sound waves]] propagate around the outer ring, guided by the channels in the circuits, which bend the waves to wrap them around the outer layers of the cloak''". This device has been described as an array of cavities which actually slow the speed of the propagating sound waves. An experimental cylinder was submerged in tank, and then it disappeared from [[sonar]]. Other objects of various shape and [[density]] were also hidden from the sonar. The acoustic cloak demonstrated effectiveness for the sound wavelengths of 40 kHz to 80 kHz.<ref name=acousticsuperlensAdler/><ref name=watery-success>
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| {{Cite web
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| | last =Laboratory News
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| | title =Watery success for Acoustic cloak
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| | publisher =Metropolis International Group Ltd,
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| | quote =Researchers from the University of Illinois – led by mechanical science and engineering professor, Nicholas Fang – have developed an acoustic cloak which renders submerged objects invisible.| url =http://www.labnews.co.uk/laboratory_article.php/6196/2/watery-success-for-acoustic-cloak-| format =Online article| accessdate =February 12, 2011}}</ref><ref name=acousticcloak>
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| {{Cite news
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| | last =Nelson
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| | first =Bryn
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| | title =New metamaterial could render submarines invisible to sonar
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| | newspaper =Defense Update
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| | format =Online
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| | date =January 19, 2011
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| | url =http://defense-update.com/wp/20110119_stealth_submarine.html
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| | accessdate =2011-01-31}}</ref><ref name=illinois>
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| {{Cite news
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| | title =Acoustic cloaking could hide objects from sonar
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| | newspaper =Information for Mechanical Science and Engineering
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| | format = Online
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| | publisher =University of Illinois (Urbana-Champaign)
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| | date =April 21, 2009
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| | url =http://mechse.illinois.edu/content/news/spotlight.php?id=177
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| | accessdate =2011-02-01}}</ref><ref>{{Cite news
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| | title =Newly Developed Cloak Hides Underwater Objects From Sonar
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| | newspaper =U.S. News - Science
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| | format = Online
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| | publisher =2011 U.S.News & World Report
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| | date =January 7, 2011
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| | url =http://mechse.illinois.edu/content/news/spotlight.php?id=177
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| | accessdate =2011-06-01}}</ref>
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| === Phononic Metamaterials for Thermal Management ===
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| As [[Phonons]] are responsible for [[Thermal conduction]] in solids, acoustic metamaterials may be designed to control heat transfer.<ref>[http://psroc.phys.ntu.edu.tw/cjp/download.php?type=full&vol=49&num=1&page=448 "Phononic Metamaterials for Thermal Management: An Atomistic Computational Study."] ''Chinese Journal of Physics'' vol. 49 , no. 1 February 2011.</ref><ref>Roman, Calvin T. [http://oai.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=ADA517273 "Investigation of Thermal Management and Metamaterials."] ''Air Force Inst. of Tech Wright-Patterson AFB OH School of Engineering and Management'', March 2010.</ref>
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| ==See also==
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| {{Multicol}}
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| *[[Acoustic dispersion]]
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| *[[Metamaterial cloaking]]
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| *[[Theories of cloaking]]
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| *[[Chirality (electromagnetism)]]
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| *[[History of metamaterials]]
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| *[[Metamaterial]]
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| *[[Metamaterial absorber]]
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| {{Multicol-break}}
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| *[[Metamaterial antennas]]
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| *[[Negative index metamaterials]]
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| *[[Nonlinear metamaterials]]
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| *[[Photonic metamaterials]]
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| *[[Photonic crystal]]
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| *[[Seismic metamaterials]]
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| *[[Split-ring resonator]]
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| {{Multicol-break}}
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| *[[Superlens]]
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| *[[Tunable metamaterials]]
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| *[[Transformation optics]]
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| ::::'''Books'''
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| *[[Metamaterials Handbook]]
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| *[[Metamaterials: Physics and Engineering Explorations]]
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| {{Multicol-break}}
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| '''Metamaterials scientists'''
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| *[[Richard W. Ziolkowski]]
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| *[[John Pendry]]
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| *[[David R. Smith]]
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| *[[Nader Engheta]]
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| *Ulf Leonhardt
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| *[[Vladimir Shalaev]]
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| {{Multicol-end}}
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| | |
| ==References==
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| {{reflist|2}}
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| | |
| ==Further reading==
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| *{{cite journal | last = Leonhardt | first = Ulf | authorlink = | coauthors= Smith, David R | year = 2008 | month = | title = Focus on Cloaking and Transformation Optics | journal = [[New Journal of Physics]] | volume = 10| issue = 11| pages = 115019| doi = 10.1088/1367-2630/10/11/115019|bibcode = 2008NJPh...10k5019L }}
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| *{{Cite journal| last1 =Fang| first1 =Nicholas| last2 =Xi| first2 =Dongjuan| last3 =Xu| first3 =Jianyi| last4 =Ambati| first4 =Muralidhar| last5 =Srituravanich| first5 =Werayut| last6 =Sun| first6 =Cheng| last7 =Zhang| first7 =Xiang| title =Ultrasonic metamaterials with negative modulus|doi=10.1038/nmat1644| pmid =16648856| year =2006| pages =452–6| issue =6| volume =5| journal =Nature Materials |url=http://xlab.me.berkeley.edu/publications/pdfs/37.%20Nick%20Nature%20Materials.pdf|bibcode = 2006NatMa...5..452F }}
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| *{{cite journal |last1=Zhang|first1=Shu|last2=Xia|first2=Chunguang|last3=Fang|first3=Nicholas|title=Broadband Acoustic Cloak for Ultrasound Waves|journal=Physical Review Letters|volume=106|year=2011|doi=10.1103/PhysRevLett.106.024301 |url=http://physics.aps.org/pdf/10.1103/PhysRevLett.106.024301.pdf|bibcode=2011PhRvL.106b4301Z}}
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| *{{cite journal |doi=10.1088/1367-2630/10/11/115032 |title=An acoustic metafluid: realizing a broadband acoustic cloak |year=2008 |last1=Pendry |first1=J B |last2=Li |first2=Jensen |journal=New Journal of Physics |volume=10 |issue=11 |pages=115032|bibcode = 2008NJPh...10k5032P }}
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| * Richard V. Craster, et al.: ''Acoustic metamaterials: negative refraction, imaging, lensing and cloaking.'' Springer, Dordrecht 2013, ISBN 978-94-007-4812-5.
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| ==External links==
| |
| *[http://www.physics.fudan.edu.cn/tps/people/jzi/research/contents/materials/Physics%20Worl%20DEC05%20thomas%20sonic%20crystals.pdf Ideas underpinning sound]
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| *[http://appliedmath.arizona.edu/modeling-computation-seminar/2009-09-17--pierre-deymier Acoustic Metamaterials and Devices: Negative, Positive, and Zero Refraction and Super-lensing in Phononic Crystals]
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| * http://iopscience.iop.org/1367-2630/10/11/115032
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| * http://news.bbc.co.uk/2/hi/science/nature/7450321.stm
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| * http://www.newscientist.com/blog/technology/2007/08/how-to-build-acoustic-invisibility.html
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| {{DEFAULTSORT:Acoustic Metamaterials}}
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| [[Category:Acoustics]]
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| [[Category:Metamaterials]]
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